The utility of separations by high performance liquid chromatography has been demonstrated over a broad range of applications including the analysis and purification of molecules ranging from low to high molecular weights. In liquid chromatography, as in gas chromatography, there are significant limitations particularly arising out of the time required for analysis. In order to understand fully these limitations, a brief description of the theoretical basis on which these separation processes are based may be useful.
The separation process relies on the fact that a number of component solute molecules in a flowing stream of a fluid percolated through a packed bed of particles, known as the stationary phase, can be efficiently separated from one another. The individual sample components are separated because each component has a different affinity for the stationary phase, leading to a different rate of migration for each component and a different exit time for each component emerging from the column. The separation efficiency is determined by the amount of spreading of the solute band as it traverses the bed or column.
The theoretical background for such separations arose in connection with the so-called "Craig machine" (as described by G. Guiochon et. al. in Fundamentals of Preparative and Non-Linear Chromatography, Academic Press (1994) at p. 174) where separations may be considered to be made in a plurality of connected, equal, discrete, hypothetical stages, each volume of which contains both stationary and moving phases and in each of which complete equilibrium is established. Each such stage is called a "theoretical" plate. In such cases the number of theoretical plates in the column is calculated from the degree of separation. The length of the column per calculated theoretical plate is called the "height equivalent to a theoretical plate" or H, and is a measure of the phenomenon of band-broadening. In chromatography, one phase is stationary and the other phase moves past the first phase at a relatively fast rate so that complete equilibrium is, in fact, not attained between the two phases, and of course, no distinct stages are observed. Notwithstanding, plate theory is commonly used to describe the passage of the solute through a chromatographic column to explain band-broadening in terms of a number of rate factors.
In applying plate theory to chromatographic columns, one must assume that all of the solute is present initially in the first plate volume of the column, the distribution coefficient is constant for the solute concentrations encountered, and the solute will rapidly distribute itself between the two phases in each plate volume.
Because columns that provide minimum broadening of separated sample bands are the sine qua non particularly of preparative, modern HPLC systems, the nature of the packing put into the column and manner in which the column is packed, all relative to the solute sought to be recovered, are of great importance. The various processes that determine relative band broadening with consequent deleterious effects on column performance are therefore desirably minimized. It was believed that the effect of each of these processes on the column plate height H can be related by rate theory to such experimental variables as mobile-phase velocity u, packing particle diameter d.sub.p, and the solute diffusion coefficient in the mobile phase. The major band broadening processes in HPLC contributing to height equivalent to a theoretical plate, H, are generally considered to be:
(1) eddy diffusion, a.sub.e d.sub.p PA1 (2) mobile-phase mass transfer, a.sub.m d.sub.p.sup.2 u/D }=Au.sup.n PA1 (3) longitudinal diffusion, bD/u=B/u, and PA1 (4) stagnant mobile-phase mass transfer, cd.sub.p.sup.2 u=Cu
where a.sub.e, a.sub.m, b, and c are plate height coefficients; n is a fractional exponent, A, B, and C are constants for a given column, and D is the diffusion coefficient of the solute in the mobile phase.
Adding these various contributions provides the classical expression describing band-spreading, well-known as the Van Deemter equation, that can be written in simplified form as: EQU H=Au.sup.n +B/u+Cu (Equ. 1)
A typical H versus u (in cm/sec) plot is shown as in FIG. 1.
In FIG. 1, the A region represents the eddy diffusion term characteristic of the column packing, i.e. flow through the column will find tortuous channels of varying length between the particles. The molecules can then travel different distances while traversing the column, resulting in band-spreading and impairing separation efficiency. Assuming that the flow profile through the column remains constant, the A term supposedly will also remain constant at all values of the linear velocity of fluid flow through the column.
The B region in FIG. 1 is a function of the linear velocity of the fluid through the column and is clearly more significant at low values of that velocity, becoming at high velocities a negligible factor contributing to band-spreading because of the process of axial molecular diffusion of the solute molecules. Such molecular diffusion is driven by concentration gradients, so the relative contribution to band-spreading becomes less as the length of time in the column becomes shorter.
The C region in FIG. 1 varies in opposite sense to B, i.e. its contribution to H increases with increasing flow rate. As the solute molecules flow faster, the separation efficiency becomes limited by the ability of the sample components to diffuse in and out of pores in the particles. The C term therefore represents the mass transfer limitation of this diffusion-driven process. For this reason, the chromatographic process exemplified by FIG. 1 has a finite time analysis boundary. Exemplary of conventional theory, the Van Deemter equation thus teaches that one must settle on a fixed analysis time to achieve maximum separation efficiency.
A family of curves similar to that of FIG. 1 can be obtained by plotting H vs. .mu. for variations in column diameter, packing particle size, amount of stationary phase and the like. According to the rate theory as exemplified by the Van Deemter equation, the minima in such family of curves indicates the optimum flow rate of the mobile phase through the column to minimize band-broadening. Importantly, the curve asserts that as the flow rate through the column increases in the C region, the column plate height H and thus the band-broadening also increases.
In describing the function of a HPLC column in a plot such as FIG. 1, it has been customary, as with the Van Deemter curves, for column plate height H to be plotted against linear velocity u of the mobile phase. Since an HPLC process is a diffusion-driven process and since different solute molecules have different diffusion coefficients, one can consider this latter variable in applying the process to a wide range of solutes of different molecular weights. Additionally, the size of the particles in the column may differ from column to column, and may also be considered as another variable, and likewise, the viscosity of the solvent for the solute might be considered. In order to normalize the plots to take these variables into account, one advantageously may employ reduced coordinates, specifically, h in place of H, and .nu. in place of u, as taught by Giddings and described in Introduction to Modern Liquid Chromatography, 2nd Ed., supra, at pp. 234-235, to yield a reduced form of the Van Deemter equation as follows: EQU h=a*+b*/.nu.+c*.nu., or (Equ. 2) EQU H=a*d.sub.p +b*D/u+c*ud.sub.p /D (Equ. 3)
wherein the coordinate h is defined as H/d.sub.p where d.sub.p is the particle diameter, and accordingly h is a dimensionless coordinate. Similarly, the dimensionless coordinate .nu. is defined as ud.sub.p /D where D is the diffusion coefficient of the solute in the mobile phase. It will be recognized that .nu. is also known as the Peclet number. It should be stressed, however, that the reduced coordinate or Peclet number, .nu., as used in the instant exposition of the present invention, is descriptive of fluid flow through the channels in the interstitial volume between particles in the column, and should not be considered as descriptive of fluid flow within the pores of porous particles constituting the packed bed in the column.
From Equ. 3, it will be seen that the eddy diffusion term, a*d.sub.p, is a function of particle size. Thus, the Van Deemter reduced equation predicts that as the particle size increases the efficiency should decrease. The longitudinal diffusion term, b*D/u, is shown as a function of both the fluid velocity and the diffusion coefficient, an inverse relationship indicating that for small molecules at very low fluid velocities, this term will be more significant. As the fluid velocity increases, this term will be dominated by the mass transfer term. The latter term, c*ud.sub.p /D, is shown as a function of all three variables, i.e. the particle size, the fluid velocity and the diffusion coefficient. As the fluid velocity increases, the equation predicts that the mass transfer term will dominate the efficiency with a deterioration proportional to the product of the velocity and particle diameter. For a given diffusion coefficient, the efficiency according to the Van Deemter equation should therefore always decrease as a function of the fluid velocity in this region. It will be shown hereinafter that these aspects of the Van Deemter equation are not valid for flow in the turbulent regime.
J. C. Giddings in Analytical Chemistry, Vol. 35, 1338, (1963), proposes that the a* term in Equ. 3 is coupled with mass transfer in the mobile phase to yield a term that is less than a*d.sub.p or c*.nu. alone. Giddings asserts that this coupling theory predicts that plate height approaches a constant value, i.e., at high flow rates the plate height is independent of flow velocity, and asserts that he has evidence of a plate height value as low as 2.5 in liquid chromatography. The same author, subsequently in Journal of Chromatography, Vol. 13, 301 (1964), writes that ". . . the value of h cannot be reduced much below 2, i.e., the plate height H cannot be pushed much below two particle diameters", presenting curves that predict that the optimum plate height is to be found at a reduced velocity between 1 and 2.
It will be appreciated that minimization of band-broadening is desirable to insure that one obtains optimum separation of solutes, particularly in analytical chromatography, and product purity, particularly in preparative chromatography. While these goals are specifically true in the separation of biological macromolecules such as industrial enzymes, various proteins for use in therapeutic and diagnostic procedures and the like, frequently such desired molecules are generated in only minute quantities in a very large volume of fluid and are very large with a correspondingly small diffusion coefficient. Thus, separation of the desired molecules by HPLC would be agonizingly slow and unduly expensive if limited to the mobile phase flow rate dictated as optimal by the Van Deemter curves. Additionally, biologicals may degrade in time while in the preparative solution, either thermally or due to the presence of proteases and the like, so speedy separation is very desirable. The efficiency of production achieved with a liquid chromatographic separation process for biological macromolecules can be described in terms of amount-of-product/dollar. To achieve optimum production, both production speed and capacity are important considerations that are currently not well met.
Efforts have been made to create HPLC systems in which separations are characterized by both high resolution and high time rate of volume processing. For example, the process disclosed and claimed in U.S. Pat. No. 5,019,270 (hereinafter the '270 patent), inter alia involves the flow of a fluid mixture of solutes through a matrix formed with two sets of interconnected pores, each set having a mean diameter that is substantially different than the other set. To effect the process, the rate of convective fluid flow, apparently under a pressure gradient, through the set of pores with the smaller mean diameter must exceed a threshold velocity that exceeds the rate of diffusion of the solute through that set of smaller pores.
Similarly, U.S. Pat. No. 5,228,989 (hereinafter the '989 patent), a continuation of a division of the '270 patent, teaches forming columns by packing particles characterized as having pore structures that are bimodal in that the particles have pores that lie within two different ranges of diameters with a specific ratio of particle diameter to the mean diameter of the pores of the larger range. U.S. Pat. No. 5,384,042 (hereinafter the '042 patent), a division of the '989 patent, discloses and claims a matrix formed essentially of the particles claimed in the '989 patent.
Notwithstanding the assertions that the teachings of the '270, '989 and '042 patents uncouples the phenomenon of band-spreading from velocity of fluid flow through a chromatographic column, it should be noted that the validity of the Van Deemter hypothesis is essentially unchallenged therein inasmuch as these patents state that the C term will not be completely independent of bed velocity and ascribe the claimed improved performance to the existence of the two related sets of pores.
Another example of pertinent prior art is set forth in Introduction to Liquid Chromatography, 2nd Ed., Snyder and Kirkland, John Wiley & Sons, N.Y. (1979), presently considered one of the authoritative textbooks on the subject, which displays at p. 238, a table 5.25 characterized as applying to "probably 99% of all LC separations for the foreseeable future. ". That table is said to dictate the conclusion that "A higher operating pressure generally yields larger N values (assuming L is increased proportionately). . . However, the advantage of a major increase in P (e.g. ten-fold as in Table 5.25) is only important for small values of d.sub.p, and/or large values of t. With small particles (5-10 .mu.m) and separation times of 15 min. to 2.5 hr., a 10-fold increase in P yields roughly a 2-fold increase in N. For much smaller particles and long separation times, a 10-fold increase in P can translate into a ten-fold increase in N. However, the experimental conditions involved are totally impractical in that separation times are much too long, and the values of d.sub.p are nonoptimum." (Where N is the number of plates, P is the pressure, d.sub.p is the particle size and L is the column length). The text continues as follows: "As separation time t increases, the optimum value of d.sub.p shifts to higher values, for example, 5 .mu.m for a separation time of 1 day. At higher operating pressures, lower values of d.sub.p are favored . . . so because D.sub.m decreases with increasing sample molecular weight, the optimum value of d.sub.p also decreases." The table on page 241 asserts that the optimum d.sub.p value for a solute having a molecular weight of 300,000 would be from about 0.03 to 0.1 .mu.m, and the text concludes with the statements "From the preceding data we see that submicron particles are decidedly advantageous for the separation of large molecules" and ". . . pressures above 5000 psi do not appear worthwhile for LC separation". (p.240). As will be apparent hereinafter, the present invention is in substantial contradistinction to these assertions and conclusions characteristic of the prior art.
A theory of chromatography expounded by M. Golay (Gas Chromatography, D. H. Desty, ed, p. 36, Buttersworths, London, 1958) is based on a number of assumptions that are correct only for laminar flow through a column. This was confirmed by J. C. Giddings (Advances in the Theory of Plate Height in Gas Chromatography, Analytical Chemistry, Vol. 35, No. 4, April 1963, pp. 439-448) who, noting shortcomings of the Van Deemter equation, asserted that the equation was incapable of successfully predicting a numerical plate height value in chromatographic columns from independent data because it contains no provision for fixing the magnitude of the effective film thickness, and contains errors and omissions relating to eddy diffusion, gas phase mass transfer and liquid film transfer.
V. Pretorius and T. Smuts (Turbulent Flow Chromatography: a New Approach to Faster Analysis, Analytical Chemistry, Vol. 38, No. 2, Feb. 1966, pp. 274-280) remark that all previous studies of minimum time required to resolve a given pair of solutes had been solely concerned with laminar flow of the mobile phase. The Pretorius et al. paper shows that if chromatography is carried out in open tubular columns with turbulent instead of laminar flow, minimum analysis time can be reduced significantly. Pretorius et al. note that under turbulent conditions the Golay expression fails, and refer particularly to the prior studies of band dispersion under turbulent conditions provided by R. Aris in Proc. Roy. Soc, A235, 67 (1956); ibid, A252, 538 (1959) to derive equations that are deemed general forms of the Golay expression supposedly valid for both laminar and turbulent flow. Pretorius et al. provide plots showing that plate height, both experimental and calculated according to the new equations, reduces dramatically on transition at a Reynolds number of about 10.sup.3 from laminar to turbulent flow through open tubes. The paper concludes that by employing turbulent rather than laminar flow, analysis times for gas chromatography are improved by about an order of magnitude. The authors speculate that for chromatography where the mobile phase is liquid, analysis time should be shortened by a factor of about 10.sup.4, based on their extrapolation of the comparison of analysis times in conventional gas and liquid chromatographies using laminar flow. Pretorius et al. also argue that the length of open tubular columns used in liquid chromatography with turbulent flow must be about ten times as long as for laminar flow, that simple separations would require pressure drops of several atmospheres, and more difficult separations imply pressure drops of more than 100 atmospheres. The use of turbulent flow liquid chromatography to obtain preparative separations with high accuracy and high speed would appear to be contraindicated by the teachings of Pretorius et al., who notes at page 280, col. 2, that a separation in a tube with a diameter of 0.1 cm. and a length of 2000 cm. (with a separation factor of 1.5) would apparently take 24 days, and surmises that with a separation factor of 0.75, the analysis time might be shortened to about 6 days.
The utility of turbulent flow in capillary chromatographic columns was advocated many years ago as an attractive means for achieving highly efficient separations, but was not extended to packed columns since the pressure drop necessary to obtain turbulent flow was considered too large for practical considerations. Cf. M. Martin et. al., Influence of Retention on Band Broadening in Turbulent Flow Liquid and Gas Chromatography., Anal. Chem., 54, 1533-1540 (1982). A similar conclusion was reached by I. Halasz in Mass Transfer in Ideal and Geometrically Deformed Open Tubes - II. Potential Applications of Ideal and Coiled Tubes in Liquid Chromatography, J. Chromatogr, 173, 229-247, (1979) which specifically declines to discuss liquid chromatography in the turbulent region because the specific permeability is at least 3 less than in laminar flow, the h values are ten times higher than theory predicts, high pressure drops are unacceptable, and injection is very difficult at high inlet pressures. Further, it was recognized that maintenance of the stationary phase in a chromatographic milieu under turbulent flow conditions was difficult at best and appeared impractical because of the high shear forces involved.
D. S. Horne et. al., A Comparison of Mobile-Phase Dispersion in Gas and Liquid Chromatography, Sep. Science, 1(5), 531-554 (1966), presented a study of the fluid dynamics of flow through columns and recognized that at very high liquid velocities, turbulence influences the rate of dispersion in the flow. The highest velocity that is shown in the liquid experiments was log v=3.8 which is approximately around a reduced velocity of 6,500. The paper states that at very high velocities and Reynolds numbers in excess of about 10, the reduced plate height becomes independent of the velocity. The paper was based on studies conducted with columns prepared with non-porous glass beads of 500 .mu.m diameter packed with column-to-particle diameter ratios of 10 to 30. No chromatographic separations were effected inasmuch as no solute was retained on the beads. Similarly, H. Kaizuma et. al., Evaluation of Coupling and Turbulence by the Dynamical Comparison of Gas and Liquid Chromatography, J. Chrom. Science, Vol. 8, 630-534 (Nov. 1970), in another study of the fluid dynamics of flow through columns packed with 500 .mu.m non-porous, spherical glass beads that were not chromatographically active, notes that coupling and turbulence will cause plate height vs. velocity plots to differ radically from the Van Deemter form, based on the use of input column pressures up to 123 atm. to get high velocity extremes. This latter paper shows that a log reduced plate height vs. log reduced velocity plot for liquids extends to reduced velocities in excess of 8,000 and shows a maximum somewhere around about 5000 before the plate heights (around 10) begin to drop. The paper concludes that turbulence does not appear to be important in LC.
With respect to the flow of an incompressible viscous fluid through a hollow cylindrical pipe or tube of uniform cross-section and relatively smooth walls, it has long been recognized that there is a critical or transition flow velocity separating steady laminar flow (i.e. where the pressure drop is proportional to the velocity) and the fluid moves in layers without irregular large fluctuations, from a regime of irregular and unsteady, or turbulent, flow (i.e. where the pressure drop varies more nearly with the square of the velocity and the local velocities and pressures in the fluid fluctuate irregularly). Such flows can be described in terms of the Reynolds number Re defined as EQU Re=.rho..nu.d/.mu. (Equ. 4)
where .rho. is the fluid density (g/cm.sup.3), .nu. is the fluid velocity (cm/sec), d is the pipe diameter (cm) and .mu. is the fluid viscosity (g/cmsec). The Reynolds number, having no dimensional units, thus serves as a criterion of the type of fluid flow. For example, it is well known that ordinarily, if the Reynolds number is small, e.g. less than about 2100, the flow in such smooth-walled tubes is laminar, and at higher Reynolds numbers, e.g. above about 3000, the flow will be turbulent. Flow at Reynolds numbers between about 2100 and 3000 constitutes the critical transition stage. The values of the Reynolds number, as noted above, depend to some extent on the smoothness of the interior surface of the conduit through which the fluid flows. Where the interior surface of the conduit is rough, i.e. irregular, the transition to turbulent flow will occur at somewhat lower Reynolds numbers. Because turbulent flow, at least in aqueous-type fluids, was largely believed to occur only if the Reynolds number exceeded about 2100, the impracticality of obtaining such flow through the packed bed of a chromatographic column seemed clear inasmuch as one cannot approach this Reynolds number except at pressures that would require massive pressure vessels and, more importantly, collapse porous particles in the column.